In terms of size limits, there's a lot of discretion in my experience.

I flew between Europe and the US this summer, in both directions with a large flat case to protect some artwork. I had checked the carrier limits for both check-in stations (*A but non-UA boarding in Europe, UA boarding in the US). My case was close enough to the limits that I certainly would not have trusted myself to eyeball the size. In truth, depending on where you made the measurements, the case was just over the limit, or just under, so I was prepared to pay the oversize fees as necessary.

Both times, no tape measures came out, and no charges were levied.